Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Near-Linear Time Computation of Welzl Orders on Graphs with Linear Neighborhood Complexity

Jan Dreier, Clemens Kuske

TL;DR

The paper presents a randomized near-linear-time algorithm for computing Welzl orders in set systems with linear primal and dual shatter functions, achieving a crossing number bound of $O(\log^2 |U|)$ with high probability. By exploiting twin and near-twin partitions and a carefully designed sampling/reconstruction scheme, the method contracts the problem size in each iteration and attains a near-linear runtime in the input size $\|S\| = |U| + \sum|X|$. It also extends to higher VC-dimension via adjustments, yielding $O(|A|^{1-1/d^2}\log^2 |A|)$ crossing numbers while maintaining $O(\|S\|\log \|S\|)$ time. The approach has concrete graph-theoretic consequences, enabling near-linear construction of compact neighborhood covers and significant improvements in first-order model checking for certain graph classes, notably monadically stable ones. Overall, the work delivers fast, probabilistic guarantees for Welzl orders in structurally well-behaved set systems, with broad applicability in computational geometry and graph theory.

Abstract

Orders with low crossing number, introduced by Welzl, are a fundamental tool in range searching and computational geometry. Recently, they have found important applications in structural graph theory: set systems with linear shatter functions correspond to graph classes with linear neighborhood complexity. For such systems, Welzl's theorem guarantees the existence of orders with only $\mathcal{O}(\log^2 n)$ crossings. A series of works has progressively improved the runtime for computing such orders, from Chazelle and Welzl's original $\mathcal{O}(|U|^3 |\mathcal{F}|)$ bound, through Har-Peled's $\mathcal{O}(|U|^2|\mathcal{F}|)$, to the recent sampling-based methods of Csikós and Mustafa. We present a randomized algorithm that computes Welzl orders for set systems with linear primal and dual shatter functions in time $\mathcal{O}(\|S\| \log \|S\|)$, where $\|S\| = |U| + \sum_{X \in \mathcal{F}} |X|$ is the size of the canonical input representation. As an application, we compute compact neighborhood covers in graph classes with (near-)linear neighborhood complexity in time \(\mathcal{O}(n \log n)\) and improve the runtime of first-order model checking on monadically stable graph classes from $\mathcal{O}(n^{5+\varepsilon})$ to $\mathcal{O}(n^{3+\varepsilon})$.

Near-Linear Time Computation of Welzl Orders on Graphs with Linear Neighborhood Complexity

TL;DR

The paper presents a randomized near-linear-time algorithm for computing Welzl orders in set systems with linear primal and dual shatter functions, achieving a crossing number bound of with high probability. By exploiting twin and near-twin partitions and a carefully designed sampling/reconstruction scheme, the method contracts the problem size in each iteration and attains a near-linear runtime in the input size . It also extends to higher VC-dimension via adjustments, yielding crossing numbers while maintaining time. The approach has concrete graph-theoretic consequences, enabling near-linear construction of compact neighborhood covers and significant improvements in first-order model checking for certain graph classes, notably monadically stable ones. Overall, the work delivers fast, probabilistic guarantees for Welzl orders in structurally well-behaved set systems, with broad applicability in computational geometry and graph theory.

Abstract

Orders with low crossing number, introduced by Welzl, are a fundamental tool in range searching and computational geometry. Recently, they have found important applications in structural graph theory: set systems with linear shatter functions correspond to graph classes with linear neighborhood complexity. For such systems, Welzl's theorem guarantees the existence of orders with only crossings. A series of works has progressively improved the runtime for computing such orders, from Chazelle and Welzl's original bound, through Har-Peled's , to the recent sampling-based methods of Csikós and Mustafa. We present a randomized algorithm that computes Welzl orders for set systems with linear primal and dual shatter functions in time , where is the size of the canonical input representation. As an application, we compute compact neighborhood covers in graph classes with (near-)linear neighborhood complexity in time \(\mathcal{O}(n \log n)\) and improve the runtime of first-order model checking on monadically stable graph classes from to .
Paper Structure (10 sections, 20 theorems, 26 equations, 3 figures)

This paper contains 10 sections, 20 theorems, 26 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Introducing the Algorithm
  4. Bounding the number of iterations
  5. Bounding the crossing number
  6. Runtime analysis
  7. Bounding the failure probability
  8. Extension for higher VC-dimensions
  9. Number of iterations for higher VC-dimensions
  10. Failure probability for higher VC-dimensions

Key Result

Theorem 1.1

Let $\mathcal{S} = (U, \mathcal{F})$ be a set system with dual shatter function $\pi^*_\mathcal{F}(k) = \mathcal{O}(k^d)$ for some $d \ge 1$. Then there exists a total order on $U$ with crossing number $\mathcal{O}(|U|^{1-1/d} \cdot \log |U|)$, and $\mathcal{O}(\log^2 |U|)$ if $d=1$.

Figures (3)

  • Figure 1: The main algorithm mentioned in \ref{['thm:main_ABE']}.
  • Figure 2: Illustration of one iteration of the algorithm. With $A = A_{\mathrm{cur}}$ being the lower and $B = B_{\mathrm{cur}}$ the upper color class
  • Figure 3: The adjusted algorithm mentioned in \ref{['thm:main2']}.

Theorems & Definitions (33)

  • Theorem 1.1: Wel88CW89Wel92
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6: ms
  • Lemma 2.1: Wel88
  • proof
  • Lemma 2.2
  • proof
  • ...and 23 more